A Study of Dufour and Soret Effect on MHD Mixed Convection Stagnation Point Flow towards a Vertical Plate in a Porous Medium

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1 International Journal of Fluids Engineering. ISSN Volume 9, Number 1 (017), pp. 1-8 International Researh Publiation House A Study of Dufour and Soret Effet on MHD Mixed Convetion Stagnation Point Flow towards a Vertial Plate in a Porous Medium Dr. Anuradha S. and Harianand N. *Professor and Head, Department of Mathematis, Hindusthan College of arts and siene, Coimbatore *Researh sholar in Department of Mathematis, Hindusthan College of arts and siene, Coimbatore Abstrat he goal of the present learning is to examine the study of dufour and soret effet on MHD mixed onvetion stagnation point flow near a vertial plate in a porous medium. he governing boundary layer equations are distorted into a set of ordinary differential equations using similarity transformations. he obtained numerial onsequenes are illustrated graphially. he skin fition profile temperature and Conentration profiles overshoot adjaent the plate on growing the hemial reation parameter, Rihardson number and magneti field parameter. 1. INRODUCION Magneto hydrodynamis is one of the signifiant sub-fields of fluid dynamis whih has pretty momentous appliations in numerous divisions of siene and equipment. When the fluid is eletrially onduting. Roman had made a detailed review on MHD in his book espeially in the field of eletrially onduting fluid. Alamet al. (006) evaluated the reading of the ombined free-fored onvetion and mass

2 Dr. Anuradha S. and Harianand N. transfer flow past a vertial porous plate in a porous medium with heat generation and thermal diffusion [1]. Anghel et al (000) deliberate and experiential dufour and soret effets on free onvetion boundary layer over a vertial surfae implanted in a porous medium []. Chiam (1994) studied two dimensional steady stagnation-point flow of an inompressible visous fluid towards a strething surfae [3].El-Arabawy (009) studied the soret and dufour effets in a vertial plate with variable surfae temperature [4].Karthikeyan et al (016) studied MHD mixed onvetion stagnation point flow towards a vertial plate embedded in a highly porous medium with heat and mass transfer in the presene of thermal radiation, internal heat generation soret and dufour effets [5].Mahapatra et al (00) investigated the steady, twodimensional stagnation-point flow of an inompressible visous fluid towards a strething surfae [6].Niranjan Hari et al (016) studied the two-dimensional steady stagnation-point flow, heat and mass transfer of an inompressible, eletrially onduting fluid through a porous medium along a vertial isothermal plate in the presene magneti field, volumetri heat generation/absorption and first order homogeneous hemial reation [7]. Postelniu (010) analyzed the effet of soret and dufour on heat and mass transfer at a stagnation point [8].Sharma et al (009) have investigated the effets of variable thermal ondutivity, heat soure/sink and variable free stream on flow of a visous inompressible eletrially onduting fluid and heat transfer on anon-onduting strething sheet in the presene of transverse magneti field near a stagnation point [9]. Sparrow et al. (1961) have disussed the temperature dependent heat soures or sinks in a stagnation point flow [10]. ak et al (010) investigated the MHD free onvetion- radiation in the presene of soret and dufour. ak et al (010) investigated the MHD free onvetion-radiation in the presene of soret and dufour [11].Vempati et al (010) studied the soret and dufour on MHD with thermal radiation [1]. Yih (1998) explored in the present paper, we have onsidered the heat soure/sink effet on MHD mixed onvetion in stagnation flows of inompressible, and eletrially onduting fluids over a vertial permeable flat plate with linear wall temperature proportional to the distane in a porous medium [13].. MAHEMAICAL MODEL Consider the two-dimensional steady stagnation-point flow, heat and mass transfer of an inompressible, eletrially onduting fluid through a porous medium along a vertial isothermal plate in the presene magneti field, volumetri heat generation/absorption and first order homogeneous hemial reation, he magneti field of onstant strength is imposed along the y-axis. he veloity distribution of the potential region flow is assumed as U x, where is a positive onstant

3 A Study of Dufour and Soret Effet on MHD Mixed Convetion Stagnation. 3 u g v Buoyany assisted region B0 Magneti field x U y w Stagnation point Buoyany opposed region B0 Magneti field Figure 1: Shemati diagram of the problem he governing equations of ontinuity of mass, momentum transfer, and Energy transfer and speies onentration are given by, u v x y 0 (1) u u u * B0 du u v v g( ) g (C C ) ( )(u U ) U x y y K dx qr C u v Q( ) D r x y y k y y u v D C S x y y y C C C 0(C ) r () (3) (4) he boundary onditions are 0, 0,,C as y o (5) u v w C u U x,, C C as y Hene the radiation effet is taken in the medium due to Stefan Boltzmann slaw whih states that the radiation is proportional to the 4 th power of temperature. Heat flux due to radiation is given by,

4 4 Dr. Anuradha S. and Harianand N. * 4 4 qr 3K y * Where and k are the Stefan-Boltzmann onstant and the mean absorptionoeffiient, respetively. Using the Rosseland approximation, it is assumed that the differenes in temperature within the flow are too small, so that, using aylor 4 series may be shown as a linear funtion of temperature about the free stream temperature, is approximately (6) (7) Equation (7) is substituted in (3) for temperature. We introduing the following dimensionless variable and parameters C C g ( ) x y xf Gr 3 0, (x, y) ( ),C r, ( ), ( ),, C C g (C C ) x Gr Gr U x 4 Gr, Ri, Ri, Re, Rd, K, S, kk K D * 3 * 3 x Re x Re x B0 Q Q Dr ( C C) Sr ( ) M, S Pr, D f, St ( ) D ( C C ) Where is the stream funtions whih is defined as the form as u, v y So x that the ontinuity equation (1) is automatially satisfied, substituting the expression in (7) together the variable in (8) into (1)-(5), we obtain the following nonlinear ordinary differential equations f ff f Ri Ri (K M)( f 1) 1 0 (9) 4 (1 Rd) Pr f S D f ( ) 0 3 (10) ( ) S f SCr S t 0 (11) (8) he orresponding boundary onditions are f 0, f 0, 1, 1 as 0 (1) f 1, 0, 0, as where Cr is a hemial reation parameter, Gr is solutal Grashof number, Gr is thermal Grashof number, K is a porous medium permeability parameter, M is

5 A Study of Dufour and Soret Effet on MHD Mixed Convetion Stagnation. 5 magneti field parameter, Pr is Prandtl number, Rd is a thermal radiation parameter, Ri is the solutal Rihardson number, Ri is the thermal Rihardson number, 3. NUMERICAL ANALYSIS In this study is to analyze dufour and soret effet on MHD mixed onvetion stagnation point flow towards a vertial plate in a porous medium. he governing boundary layer equations are transformed into a set of ordinary differential equations using similarity transformation. hen they are solved by shooting tehnique with runge kutta fourth order iteration. he obtained numerial results are illustrated graphially for different values done by using Mathematia omputer language. From the proess of numerial omputation the fluid veloity, the temperature, the onentration, the Skin frition oeffiient has analyzed. 4. RESULS AND DISCUSSION he results are displayed graphially for different parameters (K, M, Ri, Ri,S, Rd, Cr), he skin frition inreases with inreasing porous medium permeability parameter, magneti field perimeter and it dereases with inreasing the hemial reation parameter. he loal Nusselt number inreases with an inreasing permeability parameter, magneti field parameter, thermal radiation parameter and dereases with inreasing Rihardson number, heat generation parameter. he loal Sherwood number inreases with an inreasing permeability parameter, magneti field parameter, Rihardson number, internal heat generation, hemial reation parameter. From figure and 3 it has been noted that Ri and Ri be a solutal and thermal Rihardson number, It an be seen from these figures that oeffiient inrease in the Rihardson number. Figure 4 Noted that M is a magneti field parameter, with the inrease in M, dereases onsiderably owing to the a inrease in skin frition. he rate of heat transfer dereases owing to inreasing the magneti field parameter. It is observed from Figure 5 that the temperature inreases with inreasing the thermal radiation parameter. Figures 6 illustrate the temperature profiles for different values of S. We an observe that thermal boundary layer thikness is more than the momentum boundary layer thikness. Figure 7 illustrate the temperature dereases with inrease in dufour number. Figure.8 illustrates the onentration profiles for different Cr values with onstant Ri, Ri, K, M, S, Rd values. We observe that the onentration gradually dereases exponentially in the boundary layer on inreasing Cr values. Figure 9 illustrate there is an inrease in onentration with an inrease in

6 6 Dr. Anuradha S. and Harianand N. permeability of the porous medium. Figure.10 onentration inreases with inrease in soret number. his is beause either inrease in onentration differene or derease in temperature differene leads to an inrease in the value of D f resulting trends similar to the above observation. Skin frition profiles: Ri =0.1 Ri=0.5 Ri=1.0 Ri=1.5 Ri=.0 Ri=0.1 Ri=0.5 Ri=1.0 Ri=1.5 Ri=.0 M=0 M= M=4 M=6 M=8 C f C f C f Figure. Skin frition profiles for different Ri Values with Ri = 1, Ri = 0.5, K = 0.5, Rd = 0., S = 0., Cr = 0.. Figure 3. Skin frition profiles for different RiValues with Ri = 1, K = 0.5, M = 0.5,Rd = 0., S = 0., Cr = 0.. Figure 4.Skin frition profiles for different M Values with Ri = 1, Rd = 0., S = 0., Cr = 0.. emperature profiles: Rd=0 Rd=0. Rd=0.4 Rd=0.8 Rd=1.0 ( ) Figure 5.emperature profiles for differentrd values with Ri = 1, M = 0.5, S = 0., Cr = 0. Figure 6.emperature profiles for different SValueswith Ri = 1, M = 0.5, Rd = 0., Cr = 0.. Figure 7.emperature profiles for differentdf valueswith Ri = 1, M = 0.5, S = 0., Cr = 0.. Conentration profiles ( ) Cr=-1.0 Cr=-0.5 Cr=0 Cr=0.5 Cr=1.0 ( ) K=0 K=0.5 K=1.0 K=.0 ( ) Sr= Sr=4 Sr=6 Sr=8 Figure 8. Conentration profiles for different Cr values with Ri = 1, M = 0.5, Rd = 0., S = 0.. Figure 9. Conentration profiles for different K values with Ri = 1, M = 0.5, Rd = 0., S = 0.. Figure 10. Conentration profiles for differentsr Values with Ri = 1, M = 0.5, Rd = 0., S = 0..

7 A Study of Dufour and Soret Effet on MHD Mixed Convetion Stagnation. 7 CONCLUSION In this paper, we studied the learning on mass transfer effets of hemial reation on MHD mixed onvetion stagnation point flow near a vertial plate in a porous medium. he main findings are shortened as follows: he skin frition profiles overshoot neighboring the plate surfae on growing the magneti field, Rihardson number and internal heat generation parameter. he skin frition redutions on growing the porous medium permeability parameter. he temperature raises with an inrease in the heat generation and thermal radiation parameters. he onentration delines on swelling the hemial reation parameter. REFERENCES [1] Alam M. S. and M. M. Rahman, Dufour and Soret effets on mixed onvetion flow past a vertial porous flat plate with variable sution, Nonlinear Analysis: Modeling and Control, vol. 11, no. 1, pp. 3 1, 006. [] Anghel M., H. S. akhar and I. Pop, Dufour and Soret effets on free onvetion boundary layer over a vertial surfae embedded in a porous medium, Studia Universitatis Babes-Bolyai, Mathematia XLV: pp.11-,000. [3] Chiam.C, Stagnation-point flow towards a strething plate, J. Phys. So. Jpn , [4] El-Arabawy H. A. M, Soret and dufour effets on natural onvetion flow past a vertial surfae in a porous medium with variable surfae temperature, Journal of Mathematis and Statistis, vol. 5, no.3, pp , 009. [5] Karthikeyan.S, M. Bhuvaneswari, S. Sivasankaran and S. Rajan (016) Soret and Dufour Effets on MHD Mixed Convetion Heat and Mass ransfer of a Stagnation Point Flow towards a Vertial Plate in a Porous Medium with Chemial Reation, Radiation and Heat Generation journal of applied fluid mehanis, vol.9, No.3, pp ,016 [6] Mahapatra,. R. and A. S. Gupta. Heat transfer in a stagnation-point flow towards a Strething sheet. Heat and Mass ransfer 38,517-51, (00) [7] NiranjanHari, S Sivasankaran, M Bhuvaneswari and ZailanSiri Effets of hemial reation on MHD mixed onvetion stagnation point flow toward a vertial plate in a porous medium with radiation and heat generation Journal of Physis: Conferene Series 66 (016) [8] Postelniu A, Heat and mass transfer by natural onvetion at a stagnation point in a porous medium onsidering Soret and Dufour effets, Heat and Mass ransfer, vol. 46, no. 8-9, pp , 010.

8 8 Dr. Anuradha S. and Harianand N. [9] Sharma, P. R. and G. Singh Effets of variable thermal ondutivity and heat soure/sink on MHD flow near a stagnation point on a linearly strething sheet. Journal of Applied Fluid Mehanis (1), [10] Sparrow EM, CessRD emperature dependent heat soures or sinks in a stagnation point flow. Applied Siene Researh 5: [11] ak S. S., R. Mathur, R. K. Gehlot, and A. Khan, MHD free onvetionradiation interation along a vertial surfae embedded in darian porous medium in presene of soret and Dufour s effets, hermal Siene, vol. 14, no. 1, pp , 010. [1] Vempati S. R. and A. B. Laxmi-Narayana-Gari, Soret and Dufour effets on unsteady MHD flow past an infinite vertial porous plate with thermal radiation, Applied Mathematis and Mehanis, vol. 31, no. 1, pp , 010. [13] Yih K.A. Heat soure/sink effet on MHD mixed onvetion in stagnation flow on a vertial permeable plate in porous media Int. Comm. Heat Mass ransfer, Vol. 5, No. 3, pp , 1998